efﬁcientuplinkmodelingfordynamicsystem-levelsimulations...

Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2010, Article ID 282465, 15 pagesdoi:10.1155/2010/282465

Research Article

Efficient UplinkModeling for Dynamic System-Level Simulationsof Cellular andMobile Networks

Ingo Viering,1 Andreas Lobinger,2 and Szymon Stefanski3

1Nomor Research GmbH, 81541 Munich, Germany2Nokia Siemens Networks, 81541 Munich, Germany3Nokia Siemens Networks, 53-611 Wroclaw, Poland

Correspondence should be addressed to Andreas Lobinger, [email protected]

Received 11 February 2010; Accepted 23 July 2010

Academic Editor: Christian Hartmann

Copyright © 2010 Ingo Viering et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A novel theoretical framework for uplink simulations is proposed. It allows investigations which have to cover a very long (real-)time and which at the same time require a certain level of accuracy in terms of radio resource management, quality of service,and mobility. This is of particular importance for simulations of self-organizing networks. For this purpose, conventional systemlevel simulators are not suitable due to slow simulation speeds far beyond real-time. Simpler, snapshot-based tools are lackingthe aforementioned accuracy. The runtime improvements are achieved by deriving abstract theoretical models for the MAC layerbehavior. The focus in this work is long term e volution, and the most important uplink effects such as fluctuating interference,power control, power limitation, adaptive transmission bandwidth, and control channel limitations are considered. Limitations ofthe abstract models will be discussed as well. Exemplary results are given at the end to demonstrate the capability of the derivedframework.

1. Introduction

The requirements for simulation tools are changing withthe introduction of novel advanced methods. In particular,investigation of self-organizing networks (SONs) [1–5] haveto cover extremely long time intervals; however, they requirea sufficient level of accuracy in terms of radio resource man-agement (RRM), quality of service (QoS), and mobility atthe same time. For instance, self-optimization of the downtiltangle [6] is a process which may cover at least several days,since the network has to make sure that meaningful statisticson user locations and signal strengths have been collected.Furthermore, there are certainly interactions and collisionsbetween SON and RRM, so that RRM cannot be entirelyexcluded from the simulations. For instance, if the downtiltangle is changed too fast, RRM measurements might getconfused leading to an unstable system. Similar things holdfor other SON use cases such as load balancing [7], mobilityrobustness optimization, and automatic neighbor relation[5].

Typical system-level simulations [8] have a very exactimplementation of RRM and QoS by explicitly modelingall the fast decisions, typically on a millisecond time scaleor even below, for example [9]. This ends up in a verylarge simulation runtime, far beyond real-time. Simulatingseveral hours, days, or even more is impossible with thisclass of simulators. Those simulators are used to makeaccurate performance evaluations given a fixed parameterconfiguration according to specified reference scenarios.

Alternatively, the use of light, snapshot-based tools isquite popular [10, 11]. Those allow for a rapid collectionof network statistics. However, accuracy of RRM and QoS islost to a wide extent. In particular, handover effects such ashysteresis and time to trigger. can not be modeled withouthaving a true time axis implemented. Furthermore, trafficcharacteristics are poorly reflected, for example, the factthat users at the cell edge require much more resourcesthan close users in many cases. It is also more than criticalto investigate convergence behavior of dynamic SON loopswithout a real-time axis and without real mobility. Those

2 EURASIP Journal on Wireless Communications and Networking

simulators are used for network planning or for coarsestudies to understand the interrelations of new features, forexample, heterogeneous networks [12].

In this work we will present the theoretical frameworkfor a new class of simulators which is capable of making verylong SON simulations with the necessary level of accuracy.It can be understood as a smart extension of snapshot-based tools with a time axis and with abstract, semianalyticalmodels of RRM and QoS. It allows self-tuning of parametersduring the simulations (which is a typical SON aspect)rather than using a fixed parameter configuration for everysimulation. We are certainly not reaching the accuracy of fullsystem-level simulations; however, this is not needed inmanycases. For the downlink this work has already been started in[13]. Unfortunately the uplink shows a lot of fundamentaldifferences compared with the downlink which complicatesmattersin the following way.

(i) Every terminal has its own individual power budget.

(ii) The uplink typically has a power control (due tonear/far problem).

(iii) The intercell interference is heavily fluctuating.

(iv) Control channel limitations are more critical.

(v) The access scheme might be different so that thescheduling strategies are different.

Those aspects will be addressed in this work based onthe principles introduced in [13]. Although the focus of thiswork is on the introduction of the simulation framework, wewill also give some calibration results as well as some firstSON results. The derivations are based on the 3GPP standardlong-term evolution (LTE) [14]. However the principles canbe applied to other systems such as HSPA and WiMAX aswell.

We will start with definitions of the LTE uplink, theuplink power control, and the uplink SINR. In Section 3we will discuss the scheduling strategies. We will considerdifferent resource fair strategies, throughput fair strategiesand QoS strategies targeting a certain bit rate. All derivationsare done under the assumption of an adaptive transmissionbandwidth scheduler. Performance metrics are introduced inSection 4, in particular, dissatisfaction levels due to overload,power limitation, and control channel limitation. Resultswith the new framework are given in Section 5, and Section 6concludes this work. In the appendices important andinteresting properties of fairness in the uplink in comparisonto downlink fairness are discussed.

2. Definitions

We will discuss the LTE uplink, which is a Single CarrierFDMA system. [14]. The whole system bandwidth is dividedintoMtotal subbands which are called physical resource blocks(PRBs). In every transmission time interval (TTI) a user canbe assigned a subset of those Mtotal PRBs which, however,have to be adjacent. The user will spread the symbolsto transmit over this group of PRBs. Note that this so-called single carrier constraint is different to the OFDMAdownlink.

Due to the single carrier constraint a frequency selectivescheduler for the LTE uplink may have a packing problem(“Tetris” problem), that is, it might not be able to fillthe entire bandwidth in some cases. The more multiuserdiversity the scheduler aims to exploit, the larger will be thepacking problem. In this work we neglect those cutaways,that is, we assume that the scheduler can fill the entirebandwidth. Note that it is very easy to construct such ascheduler, but the frequency-selective multi-user gain will bepoor.

Random variables will be written in bold letters, forexample, v or SINR. It is very important for this work todistinguish between random and deterministic variables. Allvariables refer to linear values, except the first equations (1)to (4) that make use of the dB domain. For the sake of betternotation we are using the same symbols nevertheless.

2.1. General Definitions. We are assuming a network givenby U users u = 1 . . . U located at the coordinates −→q u, and Ccells c = 1 . . . C. All propagation effects (comprising pathloss,antenna patterns, and shadowing) between position −→q andcell c are summarized in the propagation maps Lc(

−→q ,Θc).Details on the included propagation effects are found in[13]. Note that the propagation maps are deterministic forour investigations even if the shadowing has been generatedrandomly. Fast Fading is not considered in this work.N is thethermal noise on a single PRB.

Θc is the downtilt angle of cell c. We assume that this isthe only propagation parameter which can be dynamicallyinfluenced, all others are either given by the environment(e.g., pathloss exponent, shadowing) or are configuredstatically (e.g., antenna height, azimuth orientation) and aretherefore omitted. Please note that downtilt optimization isan important SON use case, and hence we leave the downtiltangle in the equations although we do not present results onthat.

Furthermore, every cell c can adjust individual powercontrol settings given by the parameters P0c and αc accordingto [15]. We assume that user u is served by cell c = X(u),where X(u) is the connection function, and every user isconnected exactly to a single cell. In this work, we assumethat X(u) is given by the best serving cell on downlink, thatis, every user is connected to the strongest cell. This is atypical case; however we could in principle also optimize theconnection function with the equations given in this work.

The number of users in cell c is abbreviated by Nc =∑u|X(u)=c 1, and the set of users connected to cell c is

abbreviated by Uc = {u | X(u) = c}.

2.2. Power Control. Uplink Power Control is typically givenby the equation (cf. [15], neglecting the closed loop terms)

P(total)T ,u =min

(Pmax, P0X(u) + αX(u) · LX(u)

(−→q u,ΘX(u)

)

+10 · log10(Mu)),

(1)

where P(total)T ,u is the total transmit power of user u, Pmax is

the maximum transmit power, and Mu is the number of

EURASIP Journal on Wireless Communications and Networking 3

PRBs allocated to user u. In the following we will use the

transmit power per PRB P(PRB)T ,u instead of the total transmit

power P(total)T ,u . Furthermore, we assume that the scheduler at

the serving cell X(u) is smart enough that it will not driveusers into power limitation through the choice of Mu, thatis it will limit the number of PRBs Mu such that the minoperator does not expire (the min operator can only expirefor Mu = 1). This behavior will be elaborated later on inSection 3.2. In this case we can define the transmit power perPRB (actually power spectral density) as

P(PRB)T ,u = min

(Pmax,P0X(u)

+αX(u) · LX(u)(−→q u,ΘX(u)

)).

(2)

2.3. Signal-to-Noise and Interference Ratio. With this def-inition, we can write the received power of user u at itsserving cell X(u) as (we are omitting the superscript (PRB) forthe following variables although we keep on using spectraldensities/power per PRB)

PR,u = P(PRB)T ,u − LX(u)

(−→q u,ΘX(u)

). (3)

Similarly, we define the interference produced by user uat any other cell c /=X(u) as

Ic,u = P(PRB)T ,u − Lc

(−→q u,Θc

). (4)

Note that this interference is only produced if user u isscheduled by its serving cell X(u) at the time and PRB ofinterest. Let us define the random variable vc which specifiesthe user which is scheduled by cell c at a particular time anda particular PRB. We call the probability that cell c schedulesuser v the scheduling probabilities pc(v). We assume thatthe scheduling probabilities are identically distributed overtime and frequency but not independently. Correlations andfurther details of the random variables vc will be discussedlater on. As a consequence, the interference produced fromcell i to a target cell c is also a random variable:

Ic,i = Ic,vi . (5)

Furthermore the SINR for user u also gets a randomvariable (although we ignore fast fading at all):

SINRu = PR,u∑i /=X(u) IX(u),i +N

(6)

Note that whereas we have used power values in dB sofar, any power and SINR variables in this and the followingequations are linear values (using the same symbols). In thefollowing we will look at the average of this random SINR(still on a per user basis):

SINRu = Exp{SINRu}

= Exp

{PR,u∑

i /=X(u) IX(u),i +N

}

= PR,u · Exp{

1∑

i /=X(u) IX(u),i +N

}

.

(7)

Let us make some important observations.

(i) The received power PR,u is not a random variable.

(ii) The last expectation of (7) does not depend on useru, only on the cell X(u), that is, it is the same for allother users connected to cell X(u).

(iii) It is interesting to see that the more the interferenceIc,i fluctuates, the smaller gets the average SINR. Thisis easily derived from Jensen’s inequality (1/x is aconvex function).

Note that the random variable Ic,i is actually a deter-ministic function of the random variable vi (cf. (5)),that is,the interference is determined as soon as the scheduler hasselected a user vi.

2.4. Evaluation of the Expectation. Even if we already knewthe scheduling probabilities pc(v), the expectation would bevery inconvenient to evaluate. In this section, we assume thatthe scheduling probabilities are well known (we will discusslater on how to calculate them), and we will focus on theevaluation of the expectation in the average SINR expression(7). We have observed that this expectation is cell specificand does not depend on the user, so we have replaced X(u)directly by cell c:

Exp

{1

∑i /= c Ic,vi +N

}

(8)

Obviously, this expectation is multidimensional, sinceC − 1 different (independent) random variables vi’s areinvolved. We can give a closed-form expression:

∑

v1∈U1

∑

v2∈U2

· · ·∑

vC∈UC

p1(v1) · p2(v2) · · · pC(vC)∑

i /= c Ic,vi +N . (9)

Please note that cell X(u) does not contribute to theinterference on itself. However, for the sake of betterillustration we have left the corresponding sum in theequation. Unfortunately, the nested sum can hardly beevaluated numerically. For instance, in a typical scenario[16] with 57 cells and 10 users per cell we would have 1057

addends. Unfortunately, due to the nonlinearity of the 1/xfunction, there is no way to separate the random variablesand thereby the nested sums. Restricting the interferenceimpact to only close neighbors (e.g., first and second ringaround a cell) reduces the problem a bit; however it is stillhardly feasible. Note that we have used the abbreviationUc ={u | X(u) = c} which is the set of users connected to cellc.

A practical solution is a Monte Carlo integration.We generate a large number S of random C-tuples{v1,s, v2,s, . . . , vC,s} with s = 1 . . . .S containing samples ofthe random variables v1, v2, . . . , vC. As long as the numberof samples S is sufficiently large, we can get a goodapproximation of the expectation by

1S·

S∑

s=1

1∑

i /= c Ic,vi,s +N. (10)


Our investigations have shown that S ≥ 1000 gives stableresults and is still feasible from a complexity point of view.Note that for the Monte Carlo approach the generation ofthe random C-tuples certainly must follow the schedulingprobabilities p1(v1), . . . , pC(vC). Accuracy can be increasedby combining the two approaches: the first ring of interferingcells can be exactly evaluated whereas the rest of the cells isconsidered by the Monte Carlo approach. In this paper wehave only used the Monte-Carlo approach.

2.5. Rate Function. Using the previously derived SINR (perPRB) we define a rate function R(SINR) to be the data ratewhich a user can achieve on a single PRB with average SINRusing an appropriate modulation and coding scheme. Inthe simplest case we could use Shannon’s capacity equationor an extension thereof. In this work, we will follow amore realistic approach using link level results. We areusing an abstract model presented in [17] which has beenshown to be very close to real simulations using the Turbocodes defined in 3GPP [14]. The LTE uplink overheadthrough reference signals has been taken into account.Figure 1 shows the employed rate function including theShannon reference with and without considering the LTEoverhead.

Note that the Shannon bounds inherently assume a per-fect selection of modulation and coding schemes. Howeverin the uplink, due to fluctuating interference, this selectioncan not be perfect by definition, even not in static channelconditions. Furthermore imperfect channel estimation willalso degrade the performance. The consequence is a loss ofsome dBs. On the other hand, the base stations typicallyhave 2 receive antennas, which is also not considered inthe Shannon bounds which will lead to a gain in the rangeof 3 dB. Furthermore, frequency selective scheduling (e.g.,though proportional fair scheduling) will lead to multi-userdiversity gain [18, 19].

In this work we will assume that those effects willcompensate each other such that the rate function usedhere (red solid curve) is rather close to the Shannonbound considering the overhead through cyclic prefix andreference signals. Later on in Section 5.2 we will see thatthis assumption leads to a good agreement with existingsimulation results.

3. Scheduling Probabilities

Let us now have a closer look at the scheduling probabilitiespc(v). We will consider several scheduler strategies. Note thatthe random variable vc is discrete; it can adopt values v ∈ Uc

with the probability pc(v). For mathematical correctness, weneed to define a kind of idle value, for example, v = −c,with nonzero probability pc(−c) which represents the casethat no user is scheduled in cell c (at the considered timeand frequency, that is, a PRB is left empty). All other valueshave the probability pc(v) = 0.With these definitions, we canwrite (just for comprehension)

∞∑

v=−∞pc(v) = 1. (11)

−10 −5 0 5 10 15 200

200

400

600

800

1000

1200

Rate function

Shannon w/UL overheadPure Shannon bound

SINR (dB)

Through

putper

PRB(kbp

s)Figure 1: Rate function for the uplink.

3.1. General Expression. Let us define the average number ofPRBs Mu which is allocated to user u. Note that 0 ≤ Mu ≤Mtotal. Given all Mu’s in cell c, we can write the schedulingprobabilities as

pc(v) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Mv

Mtotalfor v ∈ Uc,

1−∑

u∈UcMu

Mtotalfor v = −c,

0 for otherwise.

(12)

We observe that the scheduling probabilities dependpurely on the average number of assigned PRBsMu’s. Hence,we will investigate those elaborately in the following sections.We will be looking at individual cells; we assume that cells ingeneral behave independently, that is, the random variablesvc’s are mutually independent, too.

3.2. Adaptive Transmission Bandwidth. The key differencecompared with the downlink is the fact that every user hasan individual power budget in the uplink. So we can shiftPRBs from one user to another, but not power. As a directconsequence, the maximum number of PRBs which can begiven to a user without driving it into power limitationdepends on the difference between transmit power per PRBP(PRB)T ,u (given by (2)) and the maximum transmit power Pmax

which is typically called power headroom:

Mmax,u = floor

⎛

⎝ Pmax

P(PRB)T ,u

⎞

⎠. (13)


An uplink scheduler should never assign a user morePRBs than this limit Mmax,u. Otherwise, looking at theoriginal power control equation (1), we observe that theusers would have to spread the same power over the assignedPRBs instead of increasing the power with every assignedPRB (the min operator in the PC equation (1) expires). Thisresults in an SINR loss which would eat up at least part of thebandwidth gain. Furthermore, other (non-power-limited)users can make much better use of the bandwidth. Finally,spreading the maximum power over several PRBs wouldincrease the dynamic range problems. Note that for the PCequation per PRB (2) we have already inherently assumedthat the scheduler does not exceed the aforementionedlimit. This behavior is typically called adaptive transmissionbandwidth [20].

Obviously this limits the maximum average number ofPRBs as well, since every user can be scheduled at maximum inevery time slot, hence we have

Mu ≤Mmax,u. (14)

3.3. Strict Resource Fair. The straightforward definition ofthe resource fair scheduler would be that the Nc users incell c share the available resources, that is, Mu = Mtotal/Nc.However, this may violate the power limitation of the UEs in(14). If we require resource fairness, nevertheless, that is,Mu

should be the same for all users, then every user can only getas many PRBs as the worst user (using the highest transmitpower). We can write

Mu = min

(Mtotal

NX(u), minv∈UX(u)

Mmax,v

)

. (15)

An important observation is that this solution is alsothroughput fair in the case of αc = 1 (with the exceptionthat power limited users would have smaller throughput).Otherwise (αc < 1) close users get higher throughput sincethe received power is higher and the interference is the samefor all users in a cell.

3.4. Modified Resource Fair. The previous scheduler has thedisadvantage that it may leave a lot of resources unusedalthough close users would still be able to extend theirbandwidth. Unfortunately, users at the cell edge with highpropagation loss cannot make use of the spare bandwidthdue to power limitation.

In another extreme solution we could try to always giveevery user u its maximum allowed bandwidth Mmax,u. If thisdoes not exceed the available resources, that is,

∑u∈Uc

Mu ≤Mtotal, this is a viable approach. However, this will berelatively unlikely in reality since already a single close usercould have enough transmit power to occupy more thanMtotal PRBs.

In this case we need to scale down the number of PRBs.The simplest solution would scale down all Mmax,u’s in thesame way. However this would leave too much unfairness inthe system. Instead we prefer scaling down largeMmax,u’s andbring this new solution as close as possible to the resource faircase.We will call this solution modified resource fair although

it is in general not resource fair. However, in annex A we willobserve that this solution achieves the same fairness as thetypical resource fair definition in the downlink.

We propose a simple iterative method which startswith the previous resource fair case. We define the indiceswc,1,wc,2, . . . ,wc,Nc such that they address all users u in cellc in ascending order with respect to Mmax,u’s, that is, wc,1

addresses the worst user in cell c, wc,2 addresses the secondworst user, and so forth. We will formulate our algorithm asfollows:

(1) Initialize: i = 1; M =Mtotal

(2) Abbreviate u = wc,i

(3) if M/(N − i + 1) > Mmax,u

(a) Mu =Mmax,u

(b) M = M −Mu else

(c) Mv = M/N−i+1 for all v = wc,i,wc,i+1, . . . ,wc,Nc

(d) exit

(4) Increment i = i + 1 and go to step 2

In every iteration, we check whether the remainingresource budget M equally shared among the remainingN − i + 1 exceeds the PRB limit Mmax,u of the worst ofthe remaining users u. If yes, the worst remaining usergets its maximum number of PRBs Mmax,u, and we assignthe remaining budget in the next iteration. Otherwise theremaining budget is equally shared among the remainingusers, and we exit the algorithm.

Note again that in this solution the worst user gets theleast amount of resources, but the maximum it can afford.With a high number of users this case will converge againstthe previous “Resource Fair” case.

3.5. Throughput Fair. In this section we try to approximatea throughput fair solution. We have already mentionedthat the number of PRBs is limited for the users. Sincethe interference is the same for all users the throughputachievable by all users is determined by the worst user (inparticular for α < 1). The true throughput fair solutionemploys the rate function and writes as

Mu1

Mu2= R(SINRu2)

R(SINRu1)(16)

for two users u1 and u2 in the same cell. Note thatthroughput fairness is required per cell. Unfortunately theSINRs are not known so far; recall that the Mu’s are neededto calculated scheduling probabilities and thereby the SINRs.Therefore we will give two different approximations in thefollowing.

As a first approximation, we will do the simplifyingassumption that the throughput is proportional to the SINR,that is, we assume linear rate function. From (7) we observethat the average SINR of a user within a certain cell isproportional to the received power (since the interference is


−2 0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Required

numberof

PRBs

Reference user gets 3PRBs

Linear approximationLog2 approximationReal, SINR = −6 dB

Real, SINR = −2 dBReal, SINR = 4 dBReal, SINR = 10 dB

Rx power relation PR,u2/ PR,u1 (dB)

Figure 2: Approximation of required PRBs for throughput fair case.

cell specific). In this case the throughput fair criterion of theprevious equation degenerates to

Mu1

Mu2= SINRu2

SINRu1= PR,u2

PR,u1. (17)

Another approximation which is derived from Shannon’sequation is

Mu1

Mu2= log2

(

1 +PR,u2PR,u1

)

. (18)

The comparison of the two approximations is shown inFigure 2 where we have used Mu1 = 3. The true relationobviously depends on the SINR range of the reference user(cf. legend). The linear approximation fits for very smallSINR ranges; the log2 approximation fits better for mediumSINR ranges.

Both approximations have the very nice property thatthey only depend on the positions of the users within a celland not on intercell interference or other cells in general.With those assumptions, we can formulate the throughputfair (approximated) solution in three steps.

First we assume that the worst user gets the maximumnumber of PRBs:

Mv =Mmax,v v = arguminu∈Uc

Mmax,u . (19)

Next we derive the number of PRBs for all the other users inthe cell by applying equation (17)

Mu =Mv · PR,vPR,u

, ∀u /= v (20)

or (18)

Mu =Mv · log2(

1 +PR,vPR,u

)

, ∀ u /= v. (21)

Finally we need to check whether we have exceeded theresource limit. In this case, we have to scale down all Mu’sby the same factor in order to fit into the available resourceswhilst maintaining the throughput fairness:

Mu =Mu · Mtotal

max(Mtotal,

∑u∈Uc

Mu

) . (22)

3.6. Quality of Service. A drawback of the previous methodsis that we cannot define a target QoS or a user satisfactionlevel. Inherently the methods were based on the best effortand full buffer assumption. The users always have data totransmit on one hand; on the other hand they do not have tomeet a certain target, that is, they are satisfied with whateverresourcesMu they get.

For a variety of services a certain QoS target has to bemet. For instance, users are only satisfied if they get a certainbit rate Du. If they get less, they are unsatisfied. On the otherhand, they typically cannot transmit more than Du, so thesystem will assign only the resourcesMu such that the targetrate is fulfilled, not more. Such a behavior is called constantbit rate (CBR) service.

Initially, let us assume that the SINRs are already known.We will resolve this assumption in the subsequent section.The approach is very similar to the approach in [13]. Inorder to achieve the target rateDu whilst observing the power(and therefore resource) limitation in uplink, we write therequired average number of PRBs for user u as

M(req)u = min

(

Mmax,u,Du

R(SINRu)

)

, (23)

where R(SINRu) is the rate function introduced inSection 2.5. It is important to observe that a user cannotbe satisfied if the min operator expires, irrespective of thetraffic situation in the own cell (even if the user were alone).The only way to improve those users is to decrease theintercell interference, which requires modifications in theneighboring cell such as decreasing the P0 [21]. Note thatany of those modifications is likely to reduce the QoS level inthe neighboring cell.

A cell can be defined in overload if the sum ofthe required resources exceeds the available resources,∑

u|X(u)=c M(req)u > Mtotal. In this case contention control

would drop some users (or, equivalently, admission controlwould not even have admitted some users). We assume thatthose control mechanisms work arbitrarily, that is, they donot prefer some (e.g., close) users and discriminate others(e.g., far users). This case can be modeled by applying thesame scaling procedure as in (22):

Mu =M(req)u · Mtotal

max(Mtotal,

∑u∈Uc

M(req)u

) (24)

This scaling procedure would basically make every userunsatisfied. However note that the scheduling probabilitieshere are needed to calculate SINRs. Performance metrics willbe discussed in Section 4. Alternatively, we could make use ofadmission control functionality here, which basically would


select a subset Usub,c ∈ u | X(u) = c (and drops the other

users) such that∑

u∈Usub,cM

(req)u > Mtotal is fulfilled.

We would like to emphasize again that we have assumedthat the SINRu’s are already known. However, we actuallyneed the scheduling probabilities to calculate the SINRu’sbased on (7). So in contrast to the strict resource fair,modified resource fair and (approximated) throughput fairsolutions of the previous sections, we unfortunately have notfound a closed form solution for the QoS case. This problemis very similar to the downlink problem as described in [13].

3.7. Comparison with Real-World Schedulers. In the fol-lowing we will discuss how real schedulers would mapto the previously introduced strategies. The most popularscheduler is a proportional fair (PF) scheduler. The purePF strategy is resource fair [18, 19]. However, unfortunatelythe PF definition in the uplink is not as straightforwardas it is in the downlink due to power control and powerlimitation. Most of the uplink PF strategies in LTE will useadaptive transmission bandwidth andwill be very close to themodified resource fair definition introduced in Section 3.4,when assuming full buffer/best effort traffic models (i.e.,no further QoS constraints), compare, for example, [20].Note that the scheduling gain, that is, the fact that the SINRconditioned on a user being scheduled gets better, goes intothe throughput mapping discussed in Section 2.5 and notinto the scheduling probabilities. Hence, PF and round robinstrategies are equivalent from the perspective of schedulingprobabilities (both are resource fair).

Furthermore, the PF strategies typically have to beextended with QoS constraints such as a target bit rate,minimum bit rate, or delay constraints. Those extended PFversions will come closer to the QoS scheduler describedin Section 3.6. Once again, the reduced scheduling gain(through more QoS constraints) is considered in thethroughput mapping, rather than in the scheduling proba-bilities.

3.8. Initialization of the SINRs. In this section we willpropose 2 different solutions. Let us first recall the SINRdefinition from (7)

SINRu = PR,u · Exp{· · · }. (25)

The first observation is that the abbreviated expectationExp{· · · } is only cell specific and not user specific. Hence,for a first guess of the Mu’s according to (23) and (24), weonly need to approximate a single value rather than Nc user-specific SINRu’s, which seems to be a much simpler problem.If we are applying the framework in this paper to a dynamicsimulator with a continuous time axis, we can simply takethe guess of the expectation from the previous time step.Similarly, we can read that once we know the SINRu0 of oneuser u0 (e.g., the worst user), we know all the others by thesimple relation

SINRu = SINRu0 · PR,uPR,u0

. (26)

The advantage is that it might be easier to make a guesson the SINR since it is a relative number rather than a guesson the expectation which is an absolute number. In particularthe SINR of the worst user in a cell is rather likely to be verysmall. So the second proposal is to set the SINR of the worstuser in every cell to a predefined value SINRinit (e.g., 0 dB),and the other user’s SINR in the same cell are derived fromthat according to (26). This method has the advantage that italso works with so-called snapshot-like simulators which donot have a time axis. In a dynamic simulator, this approach isprobably less accurate than the first one.

4. PerformanceMetrics

So far, we have an (almost) analytical expression SINRu forthe average SINR of every user in an LTE uplink network.Furthermore, we have already discussed the average numberMu of assigned PRBs for different scheduling strategies. Notethat in the QoS case the Mu’s actually depend on the SINRswhich are not known when calculating the Mu’s. Hence,before calculating performancemetrics we should update theMu’s with the more accurate values of the SINRs.

From these SINRu’s and Mu’s we now can start derivingseveral capacity metrics such as average cell throughput,throughput percentiles, or number of (un)satisfied users.

4.1. Throughput Metrics. In the simplest case, we calculatethe user throughputs as

Ru =Mu · R(SINRu). (27)

From those rates we can calculate a total network through-put, throughputs per cell, or throughput percentiles. Inprinciple we could also check whether users are satisfied bycomparing their data rates with the rate requirements Du’s.However recall that in (24) we have scaled down the Mu’s ofall users in case of an overload. In this case, all users wouldfall below their Du’s although in reality it might be sufficientto drop very few users to make the rest satisfied again.Furthermore, it would be interesting to have a quantitativenotion of howmuch overloaded a cell is and howmany usersare unsatisfied in fact. So for the QoS case, we will definemore appropriate performance metric in the following.

4.2. Overload and Unsatisfied Users. Exactly as in [13] wereturn to the required number of PRBs from (23) and definea virtual cell load

ρc=∑

u∈UcMu

(req)

Mtotal, (28)

which can exceed 1 thereby indicating the degree of overload.For instance, ρc = 1.1 means a 10% overloaded cell, andρc = 2 means that the cell is double overloaded, that is,half of the users will be unsatisfied. Again assuming that anadmission/contention control would exclude arbitrary users(not preferably cell edge users), we can write the number ofunsatisfied users in cell c as

Zload,c = max

(

0,Nc ·(

1− 1ρc

))

. (29)


This number accounts for dissatisfaction through overload.In addition, we will also have unsatisfied users through powerlimitation as already discussed in the context of (23), even ifthe virtual load is very small. We simply count their numberin cell

Zpower,c =∣∣∣∣

{

u ∈ Uc |Mmax,u <Du

R(SINRu)

}∣∣∣∣, (30)

where |A| returns the size of the set A. A furtherlimitation on cell level is given by the fact that the numberof users which can be scheduled at the same time isconstrained by the available resources for control channels(physical downlink control channel PDCCH in LTE). Notethat this can be a painful restriction in particular in theuplink, where the individual UE power budgets limit theability of following an aggressive TDMA strategy. Withour mathematical framework we can easily capture thislimitation as well. Assume that the maximum number ofschedulable users in cell c per TTI is given by Ktot,c. (This is asimplification. In LTE this is not a hard limit, but it dependson the user positions.) The control channel consumptionis minimized by a scheduling strategy which would alwaysassign the maximum number of resources Mmax,u accordingto (13) to a scheduled user. This maximized the numberof TTIs in which a user is not scheduled, that is, where itdoes not require any control resources. Hence, the (averaged)minimum number of required control channels required byuser u per TTI is

Ku = M(req)u

Mmax,u, (31)

using the required number of PRBs M(req)u from (23). Note

that Ku ≤ 1. Obviously, the control channels will definitely(even without any delay requirement) cause dissatisfactionin case

∑

u∈Uc

Ku > Ktot,c. (32)

Equivalent to the load dissatisfaction we will again assumethat admission/contention control would exclude arbitraryusers and thus we can define the number of unsatisfied usersdue to control channel limitation as

Zctrl,c = max

(

0,Nc ·(

1− Ktot,c∑u∈Uc

Ku

))

. (33)

Finally we have to combine the three metrics Zload,c, Zpower,c,and Zctrl,c to a single number of unsatisfied users per cell.With our high level of abstraction this is quite challengingsince the sets of load-, power-, and control-unsatisfiedusers might be overlapping. A heuristic approach wouldexclude users one by one (power-limited users first) andrecalculate the metrics until dissatisfaction has disappeared.Another approach exploits the intuitive fact that the set ofload- and control-limited users (i.e., the cell level metrics)are obviously fully overlapping. The set of power-limitedusers (user-level metric) will be rather disjoint. With those

assumptions we approximate the total number of unsatisfiedusers in cell c as

Ztotal,c = max(Zload,c,Zctrl,c

)+ Zpower,c. (34)

5. Results

A dynamic system level simulator has been implementedbased on the derivations in the previous chapters. In this sec-tion we will present some results with standard assumptions(such as full buffer traffic, proportional fair scheduler), andwe will show that those are very close to other simulationresults which have been agreed for by several companies in[9, 22]. Furthermore, we will present results with CBR traffic,and we will also look at an irregular network with SONadaptation of the power control parameters. Finally we willelaborate on the huge runtime performance.

5.1. Simulation Assumptions. We will use standard assump-tions as proposed in [16], comprising a network of 19 LTEbase stations with an intersite distance of 500m, serving57 hexagonal cells (sectors). Pathloss law, shadowing model,and horizontal beam pattern are also taken from [16], avertical pattern is not used. The users are moving with aspeed of 3 km/h, and they are handover to another cell ifthe received signal strength (measured on downlink referencesignals) with respect to the new cell is 3 dB better than thatwith respect to the serving cell (handover hysteresis). Onesimulation step is 100ms, that is, the network performanceis evaluated 10 times a second.

We are using homogeneous P0 values of P0 = −52dBmor P0 = −58dBm and a homogeneous α value of α =0.6. The resulting distribution of transmit power per PRBis shown in Figure 3. Note that this distribution does notdepend on the scheduling mechanism or traffic model sincewe record one power value for every user per simulation step.

It is obvious that the larger P0 setting of −52 dBm leadsto higher transmit powers. In this case we can also identifythe maximum transmit power of 23 dBm.

5.2. Full Buffer Traffic. We will start with the simple assump-tion of a full buffer traffic model and a modified resource fairscheduler as presented in Section 3.4. Users are uniformlydropped into the network area such that every cell serves anaverage of 10 users. The distribution of the user throughputsaccording to (27) is given in Figure 4.

As expected we observe slightly higher user throughputswith the larger P0 value. However, the difference betweenthe curves is smaller in the lower part of the plot, since thepower limitation is more critical with the smaller P0 value.The 5% percentiles (which is typically referred to as cell edgethroughput) are 420 kbps and 503 kbps whereas the averagecell throughputs are 7.3Mbps and 8.5Mbps, respectively.This is in very good agreement with the simulations in[9, 22]. The results of different companies are compared in[22] for the reference case which we have used as well. Thecell throughput results are in the range between 6.3Mbpsand 1.01Mbps, with an average of 8.6Mbps (which is alsothe result of [9]). The cell edge results span from 100 kbps to


−15 −10 −5 0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tx power (dBm)

P0 = −52 dBmP0 = −58 dBm

Cumulative

distribu

tion

function

Figure 3: Distribution of Tx Power per PRB.

0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

User throughput (kbps)

P0 = −52 dBm; average TP = 8.5MbpsP0 = −58 dBm; average TP = 7.3Mbps

Cumulative

distribu

tion

function

Figure 4: Distribution of user throughput in modified resource faircase.

460 kbps with an average of 260 kbps. Obviously our resultsare a bit too optimistic in terms of cell edge throughputwhich could be a consequence of the neglected fast fading,and, even more important, of handover gain, which isincluded in our simulations with full mobility.

5.3. Constant Bit Rate Traffic. Next we will assume a constantbit rate traffic model and a QoS scheduler as presented inSection 3.6. Different target data rates are assumed, namely,96 kbps, 256 kbps, and 512 kbps. Again, users are uniformlydropped into the network; however, the average number ofusers per cell is varied from 5 to 80. Let us first look at the

percentage of unsatisfied users due to power limitation Zpower

as given by expression (30) in Figure 5.We observe the following behavior.

(i) All curves reach a maximum and then do not growany further. The reason is that the actual load islimited and cannot exceed 100%. So the interferencewill also not grow with the number of users, and theSINRs will not decrease.

(ii) The (power) dissatisfaction level is larger for higherdata rates. This is quite obvious.

(iii) The (power) dissatisfaction level is larger for thelarger P0 = −52dBm. With smaller P0, the userscan afford more PRBs, compare (14), whereas theinterference level goes down as well (note that theother cells will reduce P0 as well in our model). Sothe SINRs remain the same as long as we do not enternoise limited regimes.

(iv) With 512 kbps and P0 = −52 we even have a“dissatisfaction floor,” that is, there will be powerlimited users even in an empty system. That is, highuplink data rates can only be supported with smallP0 values (or by relaxing the ATB power constraint(14)).

Note that the previous figure did not take into accountusers which cannot be served due to the lack of bandwidth.Figure 6 shows the total number of unsatisfied users accord-ing to (34), that is, the sum of power- and load-unsatisfiedusers. Control limitation is not considered, that is, Ktotalc =∞.

Certainly we can recognize the aforementioned dissatis-faction floor for 512 kbps and P0 = −52dBm in this figure.Otherwise, the impact of the P0 value is almost negligiblesince adding users beyond 100% virtual load obviouslymeans load-unsatisfied users hiding the aforementionedlimit for the dissatisfaction level due to the power constraint.If we target a typical overall dissatisfaction level of 5%,the uplink can satisfy 10, 21, and 56 user with 512 kbps,256 kbps, and 96 kbps, respectively. The cell throughput withthe smaller rates is around 5.4Mbps whereas the 512 kbpscase is slightly worse with 5.4Mbps due to the more criticalpower limitation.

As expected the CBR capacity is significantly below thebest effort capacity. However, the difference is smaller thanin the downlink, since the power control compensates for apart of the SINR loss of cell edge users.

5.4. Heterogeneous Scenario. Next we will leave the homoge-neous standard scenario and continue with a heterogeneousscenario with different cell sizes and nonuniform userconcentrations. Figure 7 illustrated the scenario which hasbeen proposed in [23]. The eNBs are located on an irregulargrid, 8 users are dropped into every cell, and additional 42users (i.e., 50 users in total) are dropped into cell no. 11simulating a hot spot. All users use a CBR of 64 kbps. Forevery cell c an individual P0c is chosen such that the minoperator in the power control equation (2) expires in roughly5% of the cell area.


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

Number of users per cell

Power

unsatisfied

users(%

)

P0 = −52 dBm, CBR = 96 kbpsP0 = −52 dBm, CBR = 256 kbpsP0 = −52 dBm, CBR = 512 kbpsP0 = −58 dBm, CBR = 96 kbpsP0 = −58 dBm, CBR = 256 kbpsP0 = −58 dBm, CBR = 512 kbps

Figure 5: Number of unsatisfied users due to power limitation.

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

Number of users per cell

Totalu

nsatisfied

users(%

)

P0 = −52 dBm, CBR = 96 kbpsP0 = −52 dBm, CBR = 256 kbpsP0 = −52 dBm, CBR = 512 kbpsP0 = −58 dBm, CBR = 96 kbpsP0 = −58 dBm, CBR = 256 kbpsP0 = −58 dBm, CBR = 512 kbps

Figure 6: Total number of unsatisfied users.

We will also look at load adaptive power control (LAPC)as proposed in [24] where the P0c

′s are reduced in cellswhich only carry a small load. In the CBR model reducingP0c blows up the resource consumption since the resultingSINR loss has to be compensated by bandwidth. We use avery similar approach to [24] and update the P0c(t) at time

−3000 −2000 −1000 0 1000 2000

−2000

−1500

−1000

−500

0

500

1000

1500

2000

12

3

45

678

9

1011

12 1314

15

1617

18

1920

21

2223

24

2526

27

2829

30

3132

333435

36

Distance

(m)

Distance (m)

Figure 7: Cell layout.

step t depending on the previous value P0c(t − 1) and theprevious virtual load ρc(t − 1) (note that this equation is indB scale):

P0c(t) = min

(

P0c,P0c(t − 1) + 10 log10

(ρc(t − 1)ρtarget

))

,

(35)

where ρtarget is the virtual load which we are targeting. Intheory we may want to target 100%; however, experiencehas shown that a margin should be left for handover usersso that we will use ρtarget = 80%. The rule means that weincrease the current P0c(t) if the load is above target, and wedecrease it if the load is below the target; however, we will notincrease the initial P0c which has been defined above. Notethat this automatic adaptation of a cell parameter can alreadybe considered as a SON mechanism.

Figure 8 depicts the virtual loads in the overloaded cellno.11 and its neighbors over time where we have switchedon the LAPC at t = 42 sec. Before that, the virtual loads arerather small (except the overloaded cell no.11) and differentin every cell depending on the exact position of the users andthe cell shape/size. After switching on the LAPC the virtualload in all low-loaded cells approaches the target ρtarget =80%.

The time characteristics of the corresponding P0c(t)sare shown in Figure 9. Without LAPC we can observe thatthe P0s depend on the cell size. Large cells have small P0sand vice versa (due to the aforementioned 5% rule). Afterswitching on LAPC, the low-loaded cells reduce their P0swhereas cell no.11 does not change it.

Now let us look at the impact of the LAPC on thedistribution of the interference over thermal (IoT) values.Those are based on the S samples used for the MonteCarlo approach defined in (10); the exact definition of the(instantaneous) IoT is given by

IoTc,s =∑

i /= c Ic,vi,s +N

N(36)


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Virtualload

Cell #11Cell #8Cell #9Cell #10

Cell #12Cell #30Cell #28

Time (s)

Figure 8: Virtual load in overloaded cells no.11 and neighbors.

leading to S · timesteps IoT samples per cell. Figure 10 showsthe cumulative distribution function of the IoT values (indB) in cell no.11 before and after switching on LAPC, andit also displays the linear and the harmonic averages of theIoT given by

IoTlin,c = 1S·

S∑

s=1IoTc,s,

IoTharm,c =⎛

⎝1S·

S∑

s=1

1IoTc,s

⎞

⎠

−1

.

(37)

With LAPC the CDF is steeper since the users spreadtheir data rate over a larger bandwidth leaving less PRBsunused (without interference) and leading to a smootherinterference. The interference of an individual user per PRBcertainly goes down significantly (with the P0 reduction);however, it has to occupy more PRBs to reach its CBR.Still, the linear average of the IoT is smaller with LAPCsince the rate function is concave over a wide area meaningthat decreasing the power can be compensated by a smallerincrease of the bandwidth. However, the harmonic averageshows a surprising picture. The harmonic average decreaseswith LAPC which is a consequence of the larger varianceof the IoT. Note that we have clearly shown in Section 2.3that the harmonic average is actually the relevant measure.This also manifests in the distribution of the average userSINRs defined in (7) which are shown in Figure 11 for theoverloaded cells.

The LAPC has degraded the SINR in the overloaded celleven though the P0 has not been reduced there since themore fluctuating interference obviously offers more potentialfor the link adaptation (which is assumed to be ideal inour model). This is also visible in the virtual load of the

0 20 40 60 80 100−66

−64

−62

−60

−58

−56

−54

−52

−50

−48

Cell #11Cell #8Cell #9Cell #10

Cell #12Cell #30Cell #28

Time (s)

P0(dBm)

Figure 9: P0 settings in overloaded cell no.11 and neighbors.

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Without LAPCWith LAPC

Linear average w/o LAPC

Harmonic average w/o LAPC

IoT samples (dB)

Linear average w/ LAPC

Harmonic average w/ LAPC

Cumulative

distribu

tion

function

Figure 10: IoT distribution in overloaded cell no.11 with andwithout LAPC.

overloaded cell no.11 in Figure 8 which has slightly beenincreased by LAPC (as a result of the decreased SINRs). Thisis a contrast to the results in [24] where LAPC helps toimprove the system. Fluctuating interference has a negativeimpact on link adaptation (i.e., selection of modulationand coding schemes, scheduling, etc.). In other words,smoothening the interference through LAPC will improvelink adaptation. Unfortunately this effect is not covered


−2 0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Without LAPCWith LAPC

Average SINRs (dB)

Cumulative

distribu

tion

function

Figure 11: SINR distribution in overloaded cell no.11 with andwithout LAPC.

in our simplified model, where link adaptation is alwaysassumed to be ideal. Hence, our model can exploit theaforementioned potential offered by fluctuating interference,which is not the case in reality.

Although we have gained important insights by thisanalysis, it also reveals a current limitation of the model.A remedy could be based on the principles of [25],where the rate function is elaborated by the introductionof a correlation between the SINR at the moment ofchoosing the modulation and coding scheme and themoment of applying it (where the interference mighthave changed). This correlation would increase throughLAPC.

5.5. Simulation Runtime. Finally we will look at the run-time performance of the simulation. Figure 12 shows thesimulation time for one simulation step versus the averagenumber of users per cell. It turned out that the numberof samples used for the Monte Carlo integration in (10)has significant impact. As already mentioned, convergenceis achieved for S > 1000. Fortunately we observe thatfurther reduction of this number does not bring additionalruntime benefit. The increase is linear with respect to thenumber of users, which is not surprising. With 50 users percell and a simulation step of 1 sec, which is sufficient formany applications, we are a factor 2 above real-time. Recallthat we have a fully heterogeneous 57-cell network, and nohomogeneous properties are exploited.

6. Conclusions

We have presented a very efficient modeling approach foruplink investigations focussing on the LTE standard. QoSand radio resource management (which typically work on amillisecond time scale) aremodeled in a very abstract but still

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

Average number users per cell

Runtimefor1simulation

step

500 IoT samples1000 IoT samples

5000 IoT samples10000 IoT samples

Figure 12: Runtime in a 57-cell environment on a 2.4GHz CPU.

accurate way such that the essential behavior is still included.By those means we can decrease simulation runtime farbelow real-time.

This is in particular helpful, even necessary, for simu-lations covering long time intervals. The most importantapplications are investigations on self-optimizing networkssince SON mechanisms are typically very slow control loopsand converge over hours or even days. Conventional systemlevel simulators cannot serve this purpose. On the otherhand, QoS and mobility issues are of utmost importanceand can not be neglected when studying SON which makesstatic tools inappropriate as well. We are closing the gapbetween too slow but elaborate system level simulators withfull mobility and QoS support on one side and rough staticsimulators which can lead to very fast results.

We have seen that uplink modeling is much more com-plicated than downlink modeling. The key differences arethe uplink power control (including the associated individualUE power budgets) and the multiple-access structure of theuplink (leading to extremely fluctuating interference). Wehave derived an average uplink SINR which is equivalent tothe downlink SINR which is typically intuitively used. Wehave observed that the uplink interference has to be averagedin a harmonic way.

Different traffic/scheduler assumptions have been dis-cussed. Again in contrast to downlink, there is no uniquedefinition of a resource fair scheduler in the full buffercase. We have given two solutions called strict and modifiedresource fair. Furthermore, throughput fair solutions aswell as CBR solutions targeting a given bit rate have beendefined. In order to evaluate the system performance we havediscussed uplink satisfaction in the CBR case. In additionto load limitation, we have observed that satisfaction dueto power limitation and due to control channel limitation ishighly relevant in uplink, too.


Limitations of the abstractmodels have been addressed aswell. In particular, RRM details such as the exact algorithmsfor MCS selection, multi-user diversity gains, and imperfectchannel estimation are hidden behind the abstract models(although the essence of fairness issues is considered).Furthermore, the capability to consider more elaboratetraffic models (different from pure CBR and pure best effort)is limited. We are giving an outlook on possible extensions inAppendix B.

We have given simulation results using the derivedmodeling approach. In the specified LTE test cases our resultsmatch very well the typical performance assumptions. Wehave also given results for the CBR case using differenttarget bit rates and analyzed the impact of power limitation.Finally we were looking at a heterogeneous scenario withdifferent cell sizes and non-uniform user placement. Wehave considered load-adaptive power control as an examplefor a SON mechanism. This scenario has revealed somelimitations of our modeling approach which have to beimproved in the future.

In this paper we have only looked at a small subset ofthe proposed SON use cases since the focus was on theintroduction of the framework. Certainly the model canbe used for all other SON use cases as well, such as loadbalancing, coverage and capacity optimization, or mobilityrobustness optimization.

Appendices

A. Discussion of Uplink Fairness

In this section we will compare the uplink fairness with thedownlink fairness. We will show that the modified resourcefair scheduler in the uplink achieves a comparable fairness asa typical resource fair scheduler in downlink.

In downlink, the SINRs on a PRB degrade towardsthe cell edge more severe than the pathloss law, sincethe interference grows in addition. With a resource fairscheduler, the throughputs behave accordingly. However, wecan improve cell edge users arbitrarily by assigning themmore PRBs (nonresource fair scheduling).

In the uplink, the degradation towards the cell edge isdifferent. By purely looking at the power control equation(2), bandwidth limitation (13), and SINR definition (7), wemake the following observations.

(1) The uplink interference is induced at the eNBantennas and therefore is the same for all users(as long as we do not assume intercell interferencecoordination).

(2) Assuming no power control α = 0 (resulting instrict resource fairness, that is, one PRB per user),that is, each UE transmits with maximum power,the SINRs per PRB would degrade with the pathloss.Note that this is still “fairer” as in the downlink(where interference increases in addition).

(3) Assuming power control with full pathloss compen-sation α = 1 (and adaptive transmission bandwidth),that is, all UEs are received with the same power

per PRB, the SINRs per PRB would be the same forall UEs, but the assigned bandwidth degrades withthe pathloss, copmared with (14) unless the P0s arerather small. So the throughputs degrade with thepathloss (which is a little steeper than the upper casedue to concavity of the rate function).

(4) Assuming power control with fractional pathlosscompensation 0 < α < 1 (and adaptive transmissionbandwidth), the SINRs per PRB would degrade lessseverely than the pathloss; however, the assignedbandwidth degrades with the “rest” of the pathlosslaw. So in total the throughputs again will degradewith pathloss, with the slope being in between theupper two cases.

So in either case the throughputs will degrade with thepathloss, either via SINR degradation (for α < 1) and / orvia the ATB scheduler. The degradation will be similar to thedownlink. All cases refer to the definition of the modifiedresource fair scheduler. The strict resource fair solution willlead to more fairness, that is, less throughput degradation forα > 0. For the special case of α = 1 strict resource fairnesseven leads to throughput fairness.

Despite similar fairness behavior, the uplink schedulerhas much less degrees of freedom to trade throughput amongthe users (due to individual power budgets). The only wayto extend the strict throughput limit of cell edge users isto reduce the intercell interference which unfortunately liesoutside the responsibility of the serving cell. Even moresevere, the neighbors can only decrease interference bydegrading their own users. Hence, whereas in downlink wecan trade throughputs between users, in uplink we need totrade throughputs between cells.

B. Traffic Assumptions

We have already observed that the individual power limita-tion in the uplink results in a bandwidth limitation Mmax,u

(the number of PRBs for edge users is limited) and therebyin a tight data rate limit. This is different from the downlink,where basically each PRB comes along with its own powerbudget, so that assigning more PRBs automatically meansassigning more power.

As a consequence, in the uplink we can only guaranteesmall data rates to cell edge users. So if we want to assume acommon CBRmodel for all users, it has to be very small suchthat the edge users have a fair chance to achieve it. With sucha setting there are basically 3 different methods how capacitylimit could be achieved.

(1) Common CBR. The straightforward solution is to con-sider a large number of users. This would increase simulationruntime, and large data rates would not occur at all.

(2) Common CBR Extended with Full Buffer. With a smallernumber of users we could think about distributing the excesscapacity (i.e., PRBs) among those users who still can affordmore PRBs. Basically this would be a mixture of CBR and full


buffer model, which can be referred to as guaranteed bit rate(GBR) model. The drawback is that we would need anothermetric (in addition to the satisfied users due to load andpower) accounting for users with higher throughput (i.e.,95% percentile). Furthermore we have to set a rule how theexcess capacity is distributed among the users.

(3) User-Specific CBR. Finally, we could set different bitrate requirements Du for the users, for example, dependingon the pathloss. Edge users would get small Du’s, andclose users would get higher Du’s. This might be the mostconvenient solution and probably the most realistic one aswell. However, we need to find an appropriate rule to setthe individual rate requirementsDu’s, which is absolutely notstraightforward. We will propose a simple mechanism withthe following characteristics.

(a) We define a minimum data rate Dmin which lower-bounds all rate requirements, that is, Du ≥ Dmin.

(b) The rate requirement for the worst user vc (i.e., withlargest pathloss) in every cell c is set to the minimumrate, that is,

Dvc = Dmin with vc = arguminu∈Uc

Lc(−→q ,Θc

). (B.1)

(c) The rate requirements for the other users are upscaledaccording to the pathloss relation to the worst user vc

Du = Dmin ·(LX(u)(

−→q vc,ΘX(u))

LX(u)(−→q u,ΘX(u))

)1−β. (B.2)

The fairness parameter β can be considered as “slope”of the rate requirements. β = 1 obviously means noslope, that is, the same rate Dmin for all users whichconverges to the common CBR solution. There is aninteresting relation to the power control parameterα; setting β = α will lead approximately to aresource fair behavior, since the rate increase wouldroughly be compensated by the increase in receivedpower through fractional power control; this wouldbe exactly true for a linear relation between SINR andthroughput and if we neglect the power limitation.β = 0 is the most aggressive setting which makes itapproximately equally tough for every user to achieveits target.

For this special case of user-specific CBR setting, we willnow propose a special initialization for the SINRs. Recall thatthe SINRs depend on the average number of PRBs Mu (orscheduling probabilities, resp.); however, the performancemetrics depend on the SINRs in turn. For 3 artificial caseswe have proposed simple ways to calculate the Mu’s withoutneeding the SINRs in Sections 3.3, 3.4, and 3.5. The proposalis a slight modification of the idea for the TP-fair case inSection 3.5. In every cell c we start with an initial guess G forthe worst user’s Mvc = G. From this guess we approximatethe resource consumption of

Mu = G ·(LX(u)(

−→q vc,ΘX(u))

LX(u)(−→q u,ΘX(u))

)α−β(B.3)

if we neglect power limited users. As discussed before weobserve

(i) the resource fair behaviorMu = G for α = β,

(ii) smaller resource consumptionMu < G for close userstowards the throughput fair solution α ≤ β ≤ 1,

(iii) larger resource consumption Mu > G for close userstowards the aggressive solutions 0 ≤ β ≤ α;with thissetting, we can initialize the SINR calculation for theuser-specific CBR case depending only on a singleparameter G.

(4) Summary. We will now summarize the required hooksfor the traffic configuration with user-specific CBRs.

(i) The first is a minimum rate requirement Dmin. Wepropose values between 64kbps and 96kbps. Highervalues will already become critical for cell edge users.

(ii) The second is a slope β for the rate requirements.β = 1 means throughput fair behavior; smallervalues increase the rate requirements for closer usersdepending on their pathloss relation to the worstuser, thereby forcing more load in the system. Wepropose a setting between 0 and α.

(iii) The third is a guess G for SINR initialization. A firstproposal is G = 1; however, this requires furtherstudy.

References

[1] SOCRATES, “Self-optimisation and self-configuration inwireless networks,” European Research Project, http://www.fp7-socrates.org.

[2] M. Dottling and I. Viering, “Challenges in mobile networkoperation: towards self-optimizing networks,” in Proceedingsof IEEE International Conference on Acoustics, Speech andSignal Processing (ICASSP ’09), pp. 3609–3612, Taipei, Taiwan,April 2009.

[3] M. Amirijoo, R. Litjens, K. Spaey et al., “Use cases, require-ments and assessment criteria for future self-organising radioaccess networks,” in Proceedings of the 3rd InternationalWorkshop on Self-Organizing Systems (IWSOS ’08), pp. 275–280, December 2008.

[4] “Next Generation Mobile Networks, Use Cases related toSelf Organising Network,” Overall Description, http://www.ngmn.org/.

[5] 3GPP, “Self-configuring and self-optimizing network use casesand solutions,” Tech. Rep. TR 36.902, http://www.3gpp.org/.

[6] U. Turke and M. Koonert, “Advanced site configurationtechniques for automatic UMTS radio network design,” inProceedings of the 61st IEEE Vehicular Technology Conference,vol. 3, pp. 1960–1964, Stockholm, Sweden, May-June 2005.

[7] A. Lobinger, S. Stefanski, T. Jansen, and I. Balan, “Loadbalancing in downlink LTE self-optimizing networks,” inProceedings of the 71st IEEE Vehicular Technology Conference,Taipei, Taiwan, May 2010.

[8] K. Brueninghaus, D. Astelyt, T. Salzer et al., “Link performancemodels for system level simulations of broadband radioaccess systems,” in Proceedings of the 16th IEEE International


Symposium on Personal, Indoor and Mobile Radio Communi-cations (PIMRC ’05), vol. 4, pp. 2306–2311, Berlin, Germany,September 2005.

[9] M. Rinne, M. Kuusela, E. Tuomaala et al., “A performancesummary of the evolved 3G (E-UTRA) for voice over internetand best effort traffic,” IEEE Transactions on Vehicular Technol-ogy, vol. 58, no. 7, pp. 3661–3673, 2009.

[10] A. Simonsson and A. Furuskar, “Uplink power control inLTE—overview and performance,” in Proceedings of the 68thIEEE Vehicular Technology Conference (VTC ’08), Calgary,Canada, September 2008.

[11] K. Majewski and M. Koonert, “Conservative cell load approx-imation for radio networks with shannon channels and itsapplication to LTE network planning,” in Proceedings of the6th Advanced International Conference on Telecommunications,Barcelona, Spain, May 2010.

[12] 3GPP, “Simulation assumptions and parameters for FDDHeNB RF requirements,” TDoc R4-092042, RAN WG4 Meet-ing #51, San Francisco, Calif, USA, May 2009, http://www.3gpp.org/.

[13] I. Viering, M. Dottling, and A. Lobinger, “A mathematicalperspective of self-optimizing wireless networks,” in Proceed-ings of IEEE International Conference on Communications(ICC ’09), Dresden, Germany, June 2009.

[14] 3GPP, “Stage 2 Description,” Technical Specification TS36.300, http://www.3gpp.org/.

[15] 3GPP, “Physical layer procedures,” Technical Specification TS36.213, http://www.3gpp.org/.

[16] 3GPP, “Physical layer aspects for evolved Universal Ter-restrial Radio Access (UTRA),” Tech. Rep. TR 25.814,http://www.3gpp.org/.

[17] B. Zerlin, M. Ivrlac, W. Utschick, J. Nossek, I. Viering, and A.Klein, “Joint optimization of radio parameters in HSDPA,” inProceedings of the 61st IEEE Vehicular Technology Conference(VTC ’05), vol. 1, pp. 295–299, Stockholm, Sweden, May 2005.

[18] D. Tse and P. Viswanath, Fundamentals of Wireless Commu-nications, Cambridge University Press, Cambridge, UK, 1stedition, 2005.

[19] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunisticbeamforming using dumb antennas,” IEEE Transactions onInformation Theory, vol. 48, no. 6, pp. 1277–1294, 2002.

[20] F. D. Calabrese, C. Rosa, M. Anas, P. H. Michaelsen, K.I. Pedersen, and P. E. Mogensen, “Adaptive transmissionbandwidth based packet scheduling for LTE uplink,” inProceedings of the 68th IEEE Vehicular Technology Conference(VTC ’08), Calgary, Canada, September 2008.

[21] M. Boussif, N. Quintero, F. D. Calabrese, C. Rosa, and J.Wigard, “Interference based power control performance InLTE Uplink,” in Proceedings of IEEE International SymposiumonWireless Communication Systems (ISWCS ’08), pp. 698–702,Reykjavik, Iceland, October 2008.

[22] “Next Generation Mobile Networks, Radio Performance Eval-uation,” available at, http://www.ngmn.org/.

[23] J. Turkka and A. Lobinger, “Non-regular layout for cellularnetwork system simulations,” in Proceedings of the Inter-national Symposium on Personal, Indoor and Mobile RadioCommunications (PIMRC ’10), Istanbul, Turkey, September2010.

[24] M. Boussif, C. Rosa, and J. Wigard, “Load adaptive powercontrol in LTE uplink,” in EuropeanWireless, Lucca, Italy, April2010.

[25] M. Castaneda, M. T. Ivrlac, J. A. Nossek, I. Viering, and A.Klein, “Outdated uplink adaptation due to changes in thescheduling decisions in interfering cells,” in Proceedings of

IEEE International Conference on Communications (ICC ’08),pp. 4948–4952, Beijing, China, May 2008.

efﬁcientuplinkmodelingfordynamicsystem-levelsimulations...

Documents